Fast calculation apparatus for carrying out a forward and an inverse transform

ABSTRACT

In an apparatus for carrying out a linear transform calculation on a product signal produced by multiplying a predetermined transform window function and an apparatus input signal, an FFT part ( 23 ) carries out fast Fourier transform on a processed signal produced by processing the product signal in a first processing part ( 21 ). As a result, the FFT part produces an internal signal which is representative of a result of the fast Fourier transform. A second processing part ( 22 ) processes the internal signal into a transformed signal which represents a result of the linear transform calculation. The apparatus is applicable to either of forward and inverse transform units ( 11, 12 ).

BACKGROUND OF THE INVENTION

This invention relates to a fast calculation apparatus included in eachof a forward transform calculation apparatus and an inverse transformcalculation apparatus.

A modified discrete cosine transform (hereinafter abbreviated to MDCT)apparatus is known as a linear transform apparatus for a digital signalsuch as an audio signal and a picture signal. In a conventionaltransform calculation apparatus, it is possible by the use of the MDCTtechnique to carry out a forward and an inverse transform calculationwhich are well known in the art.

The MDCT apparatus is described in detail in an article contributed byN. Schiller to the SPIE Vol. 1001 Visual Communications and ImageProcessing '88, pages 834-839, under the title of “Overlapping BlockTransform for Image Coding Preserving Equal Number of Samples andCoefficients”. The article will be described below.

In the MDCT technique, a forward transform equation and an inversetransform equation are given: $\begin{matrix}{{{y\left( {m,k} \right)} = {\sum\limits_{n = 0}^{N - 1}\quad{{x(n)}{h(n)}{\cos\quad\left\lbrack {2{\pi\left( {k + {1\text{/}2}} \right)}\left( {n + {n\quad 0}} \right)\text{/}N} \right\rbrack}}}}{and}} & (1) \\{{{{xf}\left( {m,n} \right)} = {2{f(n)}\text{/}N{\sum\limits_{k = 0}^{N - 1}\quad{{y\left( {m,k} \right)}{\cos\quad\left\lbrack {2{\pi\left( {k + {1\text{/}2}} \right)}\left( {n + {n\quad 0}} \right)\text{/}n} \right\rbrack}}}}},} & (2)\end{matrix}$wherein x represents an input signal, N represents a block length (N ismultiple of 4), m represents a block number, h represents a forwardtransform window function, f represents an inverse transform windowfunction, each of n and k represents an integer variable between 0 andN−1, both inclusive. Herein, n0 is given as follows:n0=N/4+½.  (3)

It is necessary in each of the forward and the inverse transformcalculations to carry out a large number of multiplication times andaddition times. This is because k is the integer between 0 and (N−1),both inclusive. Accordingly, an increase in the block length N resultsin an increased number of times of each of the multiplication and theaddition.

SUMMARY OF THE INVENTION

It is therefore an object of this invention to provide a forwardtransform calculation apparatus by which it is possible to reduce thenumber of times of multiplication and addition.

It is another object of this invention to provide an inverse transformcalculation apparatus by which it is possible to reduce the number oftimes of the multiplication and the addition.

It is still another object of this invention to provide a calculationapparatus in which the number of times of the multiplication and theaddition increases in proportion to only Nlog₂N, where N represents aninteger.

It is yet another object of this invention to provide a calculationapparatus in which the number of times of the multiplication and theaddition increases in proportion to (N/2)log₂(N/2), where N representsan integer.

Other object of this invention will become clear as the descriptionproceeds.

According to an aspect of this invention, there is provided an apparatusfor carrying out a forward transform calculation on an apparatus inputsignal. The apparatus includes multiplying means for multiplying apredetermined forward transform window function and the apparatus inputsignal to produce a multiplied signal and transform carrying out meansfor carrying out a linear forward transform on the product signal toproduce a forward signal representative of a result of the linearforward transform. The transform carrying out means comprises firstprocessing means connected to the multiplying means for processing theproduct signal into a processed signal, internal transform carrying outmeans connected to the first processing means for carrying out a forwardfast Fourier transform on the processed signal to produce an internalsignal representative of a result of the forward fast Fourier transform,and second processing means connected to the internal transform carryingout means for processing the internal signal into the forwardtransformed signal.

According to another aspect of this invention, there is provided anapparatus for carrying out an inverse transform calculation on anapparatus input signal. The apparatus includes transform carrying outmeans for carrying out a linear inverse transform on the apparatus inputsignal to produce an inverse signal representative of a result of thelinear inverse transform and multiplying means for multiplying apredetermined inverse transform window function and the inverse signalto produce a product signal. The transform carrying out means comprisesfirst processing means for processing the apparatus input signal into aprocessed signal, internal transform carrying out means connected to thefirst processing means for carrying out an inverse fast Fouriertransform on the processed signal to produce an internal signalrepresentative of a result of the inverse fast Fourier transform, andsecond processing means connected to the internal transform carrying outmeans for processing the internal signal into the inverse transformedsignal.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 is a block diagram of a conventional calculation apparatus forsuccessively carrying out a forward and an inverse transform;

FIG. 2 is a block diagram of a calculation apparatus according to afirst embodiment of this invention;

FIG. 3 is a flow chart for use in describing operation of a firstforward processing part included in the calculation apparatus of FIG. 2;

FIG. 4 is a block diagram of a calculation apparatus according to asecond embodiment of this invention;

FIG. 5 is a flow chart for use in describing operation of a firstforward processing part included in the calculation apparatus of FIG. 4;

FIG. 6 is a flow chart for use in describing operation of a firstinverse processing part included in the calculation apparatus of FIG. 4;and

FIG. 7 is a flow chart for use in describing operation of a secondinverse processing part included in the calculation apparatus of FIG. 4.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

Referring to FIG. 1, a conventional calculation apparatus will bedescribed at first for a better understanding of the present invention.The conventional calculation apparatus is for successively carrying outa forward and an inverse transform and comprises forward and inversetransform units 11 and 12.

The forward transform unit 11 comprises an input buffer part 13, aforward transform window part 14, and a forward calculation part 15. Theinput buffer part 13 is for memorizing N samples of original signalsx(n), as an original data block. This means that the original data blockhas an original block length N. Responsive to the original signals x(n),the forward transform window part 14 carries out multiplication betweeneach of the original signals x(n) and the forward transform windowfunction h(n) to produce a product signal xh(n) as follows:xh(n)=x(n)h(n).  (4)Responsive to the product signal xh(n), the forward calculation part 15calculates the left-hand side of Equation (1) as follows:$\begin{matrix}{{y\left( {m,k} \right)} = {\sum\limits_{n = 0}^{N - 1}\quad{{{xh}(n)}{{\cos\quad\left\lbrack {2{\pi\left( {k + {1\text{/}2}} \right)}\left( {n + {n\quad 0}} \right)\text{/}N} \right\rbrack}.}}}} & (5)\end{matrix}$The calculation part produces the left-hand side as a forwardtransformed signal y(m, k). It is necessary to carry out multiplicationN² times and addition N(N−1) times. This is because k is variablebetween 0 and (N−1), both inclusive. Depending on the circumstances,each sample of the original signals x(n) is herein called an apparatusinput signal.

The inverse transform unit 12 comprises an inverse calculation part 16,an inverse transform window part 17, and an output buffer part 18. Theinverse calculation part 16 calculates the left-hand side of Equation(2) as follows: $\begin{matrix}{{{xt}\left( {m,n} \right)} = {2\text{/}N{\sum\limits_{k = 0}^{N - 1}\quad{{y\left( {m,k} \right)}{{\cos\quad\left\lbrack {2{\pi\left( {k + {1\text{/}2}} \right)}\left( {n + {n\quad 0}} \right)\text{/}n} \right\rbrack}.}}}}} & (6)\end{matrix}$The calculation part 16 produces the left-hand side as an inversetransformed signal xt(m, n). The inverse transform window part 17multiplies the inverse transformed signal xt(m, n) and an inversetransform window function f(n) into a product in accordance with:xf(m, n)=xt(m, n)f(n).  (7)The inverse transform window part 17 thereby produces a product signalxf(m, n) representative of the product.

The product signal is supplied to the output buffer part 18 whenever themultiplication is carried out by the inverse transform window part 17.As a result, the output buffer part 18 memorizes a plurality of theproduct signals as a current and a previous data block at a time. Thecurrent data block corresponds to the original data block. The previousdata block is previous to the current data block. Each of the currentand the previous data blocks is divided into a former and a latter halfblock. The former half block comprises zeroth through (N/2−1)-th productsignals. The latter half block comprises N/2-th through (N−1)-th productsignals.

As will be understood from the following equation, the output bufferpart 18 carries out addition between the product signal xf(m, n) of theformer half block of the current data block and the product signalxf(m−1, n) of the latter half block of the previous data block toproduce a modified or reproduced signal x′(n) of a modified half blockhaving a modified block length which is a half of the original blocklength. The following equation is:x′(n)=xf(m−1, n+N/2)+xf(m, n),  (8)when 0≦n<N/2−1.Simultaneously, the output buffer part 18 memorizes, as the latter halfblock, the second product signal xf(m, n) of the current data block.Each of the forward and the inverse transform window functions h(n) andf(n) can be given by Equation (9) on page 836 in the above-mentionedarticle.

The description will now proceed to an example of an algorithm which isapplicable to a forward transform calculation apparatus according tothis invention. Substituting Equation (3), Equation (5) is rewritteninto: $\begin{matrix}\begin{matrix}{{y\left( {m,k} \right)} = {\sum\limits_{n = 0}^{N - 1}\quad{{{xh}(n)}{\cos\quad\left\lbrack {2{\pi\left( {k + {1\text{/}2}} \right)}\left( {n + {n\quad 0}} \right)\text{/}N} \right\rbrack}}}} \\{= {\sum\limits_{n = 0}^{N - 1}\quad{{{xh}(n)}{\cos\quad\left\lbrack {2{\pi\left( {k + {1\text{/}2}} \right)}\left( {n + {N\text{/}4} + {1\text{/}2}} \right)\text{/}N} \right\rbrack}}}} \\{= {\sum\limits_{n = {N\text{/}4}}^{{5N\text{/}4} - 1}{{{xh}\left( {n - {N\text{/}4}} \right)}{\cos\left\lbrack {2{\pi\left( {k + {1\text{/}2}} \right)}\left( {n + {1\text{/}2}} \right)\text{/}N} \right\rbrack}}}}\end{matrix} & (9)\end{matrix}$

In Equation (9), the cosine has a nature such that: $\begin{matrix}{{\cos\quad\left\lbrack {2{\pi\left( {k + {1\text{/}2}} \right)}\left( {n + {1\text{/}2}} \right)\text{/}N} \right\rbrack} = {{\cos\quad\left\lbrack {2{\pi\left( {k + {1\text{/}2}} \right)}\left( {\left( {n - N} \right) + {1\text{/}2} + N} \right)\text{/}N} \right\rbrack} = {{\cos\quad\left\lbrack {{2{\pi\left( {k + {1\text{/}2}} \right)}\left( {\left( {n - N} \right) + {1\text{/}2}} \right)\text{/}N} + {2{\pi\left( {k + {1\text{/}2}} \right)}N\text{/}N}} \right\rbrack} = {- {\cos\quad\left\lbrack {2{\pi\left( {k + {1\text{/}2}} \right)}\left( {\left( {n - N} \right) + {1\text{/}2}} \right)\text{/}N} \right\rbrack}}}}} & (10)\end{matrix}$

When n is shifted by (−N), the cosine has an argument shifted by2π(k+½), namely, an odd integral multiple of π. In this event, thecosine has an absolute value unchanged and a sign inverted betweenpositive and negative. Therefore, it is possible to delete the term N/4in the argument of xh in the right-hand side of Equation (9) by shiftingn by (−N) with the sign of the product signal xh(n) inverted.

Herein, the product signal is processed into particular and specificdatum x2(n) which are represented as follows:x2(n)=−xh(n+3/N4),  (11a)when 0≦n<N/4andx2(n)=xh(n−N/4),  (11b)when N/4≦n<N.

Therefore: $\begin{matrix}{{{y\left( {m,k} \right)} = {{\sum\limits_{n = 0}^{N - 1}\quad{{{x2}(n)}{\cos\left\lbrack {2{\pi\left( {k + {1\text{/}2}} \right)}\left( {n + {1\text{/}2}} \right)\text{/}N} \right\rbrack}}}\quad = {{{real}\left\lbrack {\sum\limits_{n = 0}^{N - 1}{{{x2}(n)}{\exp\left( {{- 2}\pi\quad{j\left( {k + {1\text{/}2}} \right)}\left( {n + {1\text{/}2}} \right)\text{/}N} \right)}}} \right\rbrack}\quad = {{{real}\left\lbrack {{\sum\limits_{n = 0}^{N - 1}{{{x2}(n)} \times {\exp\left( {{{- 2}\pi\quad{j\left( {k + {1\text{/}2}} \right)}\text{/}2N} - {2\pi\quad{j\left( {k + {1\text{/}2}} \right)}n\text{/}N}} \right)}}},} \right\rbrack}\quad\quad = {{{real}\left\lbrack {{\exp\left( {{- \pi}\quad 2{j\left( {k + {1\text{/}2}} \right)}\text{/}2N} \right)} \times {\sum\limits_{n = 0}^{N - 1}{{{x2}(n)}{\exp\left( {{- 2}\pi\quad{j\left( {k + {1\text{/}2}} \right)}n\text{/}N} \right)}}}} \right\rbrack}\quad = {{{real}\left\lbrack {{\exp\left( {{- 2}\pi\quad{j\left( {k + {1\text{/}2}} \right)}\text{/}2N} \right)} \times {\sum\limits_{n = 0}^{N - 1}{{{x2}(n)}{\exp\left( {{{- 2}\pi\quad{jn}\text{/}2N} - {2\pi\quad{jkn}\text{/}N}} \right)}}}} \right\rbrack}\quad = {{real}\left\lbrack {{\exp\left( {{- 2}\pi\quad{j\left( {k + {1\text{/}2}} \right)}\text{/}2N} \right)} \times {\sum\limits_{n = 0}^{N - 1}{{{x2}(n)}{\exp\left( {{- 2}\pi\quad{jn}\text{/}2N} \right)}{\exp\left( {{- 2}\pi\quad{jkn}\text{/}N} \right)}}}} \right\rbrack}}}}}}},} & (12)\end{matrix}$

where j represents an imaginary unit.

It will be assumed that:x3(n)=x2(n)exp(−2πjn/2N).  (13)

In this event: $\begin{matrix}{{y\left( {m,k} \right)} = {{{real}\left\lbrack {{\exp\left( {{- 2}\pi\quad{j\left( {k + {1\text{/}2}} \right)}\text{/}2N} \right)} \times {\sum\limits_{n = 0}^{N - 1}{{{x3}(n)}{\exp\left( {{- 2}\pi\quad{jkn}\text{/}N} \right)}}}} \right\rbrack}.}} & (14)\end{matrix}$

Herein, Equation (14) is divided into a front part and a rear part whichrepresents execution of a fast Fourier transform (hereinafterabbreviated to FFT) at a point N of a specific signal x3(n). The FFT isrepresented by Equation (15) which is given below.

Accordingly, Equation (16) is given from Equation (14). $\begin{matrix}{{{{xf}\left( {m,k} \right)} = {\sum\limits_{n = 0}^{N - 1}\quad{{{x3}(n)}{\exp\left( {{- 2}\pi\quad{jkn}\text{/}N} \right)}}}}{and}} & (15) \\{{y\left( {m,k} \right)} = {{{real}\left\lbrack {{\exp\left( {{- 2}\pi\quad{j\left( {k + {1\text{/}2}} \right)}\text{/}2N} \right)}{{xf}\left( {m,k} \right)}} \right\rbrack}.}} & (16)\end{matrix}$It is therefore possible to obtain the result of Equation (5).

The multiplications are required N times before execution of the FFT, atmost Nlog₂N times for executing the FFT, and N times after execution ofthe FFT. As a result, the multiplication is carried out the number oftimes which is substantially equal to Nlog₂N if N is great. The numberof times of the addition is equal to 2Nlog₂N. Accordingly, it ispossible to reduce the number of times of the multiplication and theaddition as compared with the conventional calculation apparatus.

The description will proceed to an example of an algorithm which isapplicable to an inverse transform calculation apparatus according tothis invention. Substituting Equation (3), Equation (6) is rewritteninto: $\begin{matrix}{{{xt}\left( {m,n} \right)} = {{2\text{/}N{\sum\limits_{k = 0}^{N - 1}{{y\left( {m,k} \right)}{\cos\left\lbrack {2{\pi\left( {n + {n\quad 0}} \right)}\left( {k + {1\text{/}2}} \right)\text{/}N} \right\rbrack}}}}\quad = {{2\text{/}N{\sum\limits_{k = 0}^{N - 1}{{y\left( {m,k} \right)}{\cos\left\lbrack {2{\pi\left( {n + {N\text{/}4} + {1\text{/}2}} \right)}\left( {k + {1\text{/}2}} \right)\text{/}N} \right\rbrack}}}}\quad = {{2\text{/}{{N{real}}\left\lbrack {\sum\limits_{k = 0}^{N - 1}{{y\left( {m,k} \right)} \times {\exp\left( {2\pi\quad{j\left( {n + {N\text{/}4} + {1\text{/}2}} \right)}\left( {k + {1\text{/}2}} \right)\text{/}N} \right)}m}} \right\rbrack}}\quad\quad = {{2\text{/}{{N{real}}\left\lbrack {\sum\limits_{k = 0}^{N - 1}{{y\left( {m,k} \right)} \times {\exp\left( \quad{{2\pi\quad{j\left( {n + {N\text{/}4} + {1\text{/}2}} \right)}\text{/}2N} + {2\pi\quad{j\left( {n + {N\text{/}4} + {1\text{/}2}} \right)}k\text{/}N}} \right)}}} \right\rbrack}}\quad = {{2\text{/}{{N{real}}\left\lbrack {{\exp\left( {2\pi\quad{j\left( {n + {N\text{/}4} + {1\text{/}2}} \right)}\text{/}2N} \right)} \times {\sum\limits_{k = 0}^{N - 1}{{y\left( {m,k} \right)}{\exp\left( {2\pi\quad{j\left( {n + {N\text{/}4} + {1\text{/}2}} \right)}k\text{/}N} \right)}}}} \right\rbrack}}\quad = {{2\text{/}{{N{real}}\left\lbrack {{\exp\left( {2\pi\quad{j\left( {n + {N\text{/}4} + {1\text{/}2}} \right)}\text{/}2N} \right)} \times {\sum\limits_{k = 0}^{N - 1}{{y\left( {m,k} \right)}{\exp\left( {{2\pi\quad{j\left( {{N\text{/}4} + {1\text{/}2}} \right)}k\text{/}N} + {2\pi\quad{{jn}k}\text{/}N}} \right)}}}} \right\rbrack}}\quad = {2\text{/}N\quad{{{real}\left\lbrack {{\exp\left( {2\pi\quad{j\left( {n + {N\text{/}4} + {1\text{/}2}} \right)}\text{/}2N} \right)} \times \quad{\sum\limits_{k = 0}^{N - 1}{{y\left( {m,k} \right)}\quad{\exp\left( {2\quad\pi\quad{j\left( {{N\text{/}4} + {1\text{/}2}} \right)}k\text{/}N} \right)}{\exp\left( {2\pi\quad{{jn}k}\text{/}N} \right)}}}} \right\rbrack}.}}}}}}}}} & (17)\end{matrix}$

The following equations are introduced:y2(m, k)=y(m, k)exp[2πj(N/4+1/2)k/N]  (18)

and $\begin{matrix}{{y\quad 3\left( {m,k} \right)} = {\sum\limits_{k = 0}^{{N\text{/}2} - 1}\quad{y\quad 2\left( {m,k} \right){{\exp\left( {2\pi\quad{jnk}\text{/}N} \right)}.}}}} & (19)\end{matrix}$

Equation (19) represents the inverse FFT for the signal y2(m, k).Substituting Equations (18) and (19) in Equation (17):xt(m, n)=2/Nreal[exp(2πj(n+N/4+1/2)/2N)y3(m, n)].  (20)It is therefore possible to obtain the result of Equation (6).

The multiplication is carried out the number of times which issubstantially equal to Nlog₂N. The number of times of the addition isequal to 2Nlog₂N. Accordingly, it is possible to reduce the number oftimes of the multiplication and the addition as compared with theconventional apparatus.

The description will now proceed to another example of the algorithmthat is applicable to the forward transform calculation apparatusaccording to this invention. With p substituted for n, Equation (9) isrewritten into: $\begin{matrix}{{y\left( {m,k} \right)} = {\sum\limits_{p = {N\text{/}4}}^{{5N\text{/}4} - 1}{{{xh}\left( {p - {N\text{/}4}} \right)}{{\cos\quad\left\lbrack {2{\pi\left( {k + {1\text{/}2}} \right)}\left( {p + {1\text{/}2}} \right)\text{/}N} \right\rbrack}.}}}} & (21)\end{matrix}$It is possible to delete the term N/4 in the argument of xh in theright-hand side of Equation (21) by shifting p by (−N) with the sign ofthe product signal xh(p) inverted.

Herein, the product signal is processed into the particular and thespecific data x2(p) which are represented as follows:x2(p)=−xh(p+3N/4),  (22a)when 0≦p<N/4andx2(p)=xh(p+N/4),  (22b)when N/4≦p<N.

Therefore: $\begin{matrix}{{Therefore}\text{:}} & \quad \\{{y\left( {m,k} \right)} = {\sum\limits_{p = 0}^{N - 1}{{{x2}(p)}{\cos\quad\left\lbrack {2{\pi\left( {k + {1\text{/}2}} \right)}\left( {p + {1\text{/}2}} \right)\text{/}N} \right\rbrack}}}} & (23)\end{matrix}$

In Equation (23), the cosine has a nature such that: $\begin{matrix}{{\cos\quad\left\lbrack {2{\pi\left( {k + {1\text{/}2}} \right)}\left( {p + {1\text{/}2}} \right)\text{/}N} \right\rbrack} = {{\cos\quad\left\lbrack {{- 2}{\pi\left( {k + {1\text{/}2}} \right)}\left( {p + {1\text{/}2}} \right)\text{/}N} \right\rbrack} = {{\cos\quad\left\lbrack {2{\pi\left( {k + {1\text{/}2}} \right)}\left( {\left( {N - p - {1\text{/}2}} \right) - N} \right)\text{/}N} \right\rbrack} = {\quad{\cos\quad\left\lbrack {{{2{\pi\left( {k + {1\text{/}2}} \right)}\left( {\left( {N - 1 - p} \right) + {1\text{/}2}} \right)\text{/}N} - \quad{2{\pi\left( {k + {1\text{/}2}} \right)}\left( {N\text{/}N} \right\rbrack}} = {- {\cos\quad\left\lbrack {2{\pi\left( {k + {1\text{/}2}} \right)}\left( {\left( {N - p - 1} \right) + {1\text{/}2}} \right)\text{/}N} \right\rbrack}}} \right.}}}}} & (24)\end{matrix}$

When (N−1−p) is substituted for p, the cosine has an absolute valueunchanged and a sign inverted between positive and negative. When pseparated into an even and an odd number, Equation (23) is rewritteninto: $\begin{matrix}{{y\left( {m,k} \right)} = {{{\sum\limits_{p = 0}^{{N\text{/}2} - 1}{{{x2}\left( {2p} \right)}{\cos\quad\left\lbrack {2{\pi\left( {k + {1\text{/}2}} \right)}\left( {{2p} + {1\text{/}2}} \right)\text{/}N} \right\rbrack}}} + {\sum\limits_{p = 0}^{{N\text{/}2} - 1}{{{x2}\left( {N - 1 - {2p}} \right)}{\cos\quad\left\lbrack {2{\pi\left( {k + {1\text{/}2}} \right)} \times \left( {N - 1 - {2p} + {1\text{/}2}} \right)\text{/}N} \right\rbrack}}}}\quad = {\sum\limits_{p = 0}^{{N\text{/}2} - 1}{\left\lbrack {{{x2}\left( {2p} \right)} - {{x2}\left( {p - 1 - {2p}} \right)}} \right\rbrack \times {\cos\quad\left\lbrack {2{\pi\left( {k + {1\text{/}2}} \right)}\left( {{2p} + {1\text{/}2}} \right)\text{/}N} \right\rbrack}}}}} & (25)\end{matrix}$

It is assumed that:x3(p)=x2(2p)−x2(N−1−2p),  (26)when 0≦p≦N/2−1.

In this event: $\begin{matrix}{{y\left( {m,k} \right)} = {{\sum\limits_{n = 0}^{{N\text{/}2} - 1}{{{x3}(n)}{\cos\quad\left\lbrack {2{\pi\left( {k + {1\text{/}2}} \right)}\left( {{2n} + {1\text{/}2}} \right)\text{/}n} \right\rbrack}}}\quad = \quad{{r\quad e\quad a\quad{l\left\lbrack {\sum\limits_{n = 0}^{{N\text{/}2} - 1}{{{x3}(n)}{\exp\left( {{- 2}\pi\quad{j\left( {k + {1\text{/}2}} \right)}\left( {{2n} + {1\text{/}2}} \right)\text{/}N} \right)}}} \right\rbrack}}\quad = {{{real}\left\lbrack {\sum\limits_{n = 0}^{{N\text{/}2} - 1}\quad{{{x3}(n)} \times {\exp\left( {{{- 2}\pi\quad{j\left( {k + {1\text{/}2}} \right)}\text{/}2N} - {2\pi\quad{j\left( {k + {1\text{/}2}} \right)}2n\text{/}2}} \right)}}} \right\rbrack}\quad\quad = {{{real}\left\lbrack {{\exp\left( {{- 2}\pi\quad{j\left( {k + {1\text{/}2}} \right)}\text{/}2N} \right)} \times {\sum\limits_{n = 0}^{{N\text{/}2} - 1}\quad{{{x3}(n)}{\exp\left( {{- 2}\pi\quad{j\left( {k + 1} \right)}n\text{/}N} \right)}}}} \right\rbrack}\quad = {{{real}\left\lbrack {{\exp\left( {{- 2}\pi\quad{j\left( {k + {1\text{/}2}} \right)}\text{/}2N} \right)} \times {\sum\limits_{n = 0}^{{N\text{/}2} - 1}\quad{{{x3}(n)}{\exp\left( {{{- 2}\pi\quad{jn}\text{/}N} - {2\pi\quad{jk2n}\text{/}N}} \right)}}}} \right\rbrack}\text{}\quad = {{real}\left\lbrack {{\exp\left( {{- 2}\pi\quad{j\left( {k + {1\text{/}2}} \right)}\text{/}2N} \right)} \times {\sum\limits_{n = 0}^{{N\text{/}2} - 1}\quad{{{x3}(n)}{\exp\left( {{- 2}\pi\quad{jn}\text{/}N} \right)}{\exp\left( {{- 2}\pi\quad{jkn}\text{/}\left( {N\text{/}2} \right)} \right)}}}} \right\rbrack}}}}}}} & (27)\end{matrix}$

It is assumed that:x4(p)=x3(p)exp(−2πjp/N).  (28)

In this event: $\begin{matrix}{{y\left( {m,k} \right)} = {{real}\left\lbrack \quad{{\exp\left( {{- 2}\pi\quad{j\left( {k + {1\text{/}2}} \right)}\text{/}2N} \right)} \times {\sum\limits_{p = 0}^{{N\text{/}2} - 1}\quad{{{x4}(p)}{\exp\left( {{- 2}\pi\quad{jkp}\text{/}\left( {N\text{/}2} \right)} \right)}}}} \right\rbrack}} & (29)\end{matrix}$

Herein, Equation (14) is divided into a front part and a rear part whichrepresents execution of the FFT at N/2 of x4(p). The FFT is representedby Equation (30) which is given below. Accordingly, Equation (31) isgiven from Equation (29). $\begin{matrix}{{{xf}\left( {m,k} \right)} = {\sum\limits_{p = 0}^{{N\text{/}2} - 1}{{{x4}(p)}{\exp\left\lbrack {{- 2}\pi\quad{jkp}\text{/}\left( {N\text{/}2} \right)} \right\rbrack}}}} & (30)\end{matrix}$  y(m, k)=real[exp(−2πj(k+1/2)/2N)×f(m, k)]  (31)It is therefore possible to obtain the result of Equation (5).

The multiplication is carried out the number of times which issubstantially equal to (N/2) log₂ (N/2) if N is great. The total numberof the addition and the subtraction is equal to N/2 for x3(p) and toNlog₂(N/2) for FFT. The number of the addition is substantially equal toNlog₂(N/2) if N is great. Accordingly, it is possible to reduce thenumber of times of the multiplication and the addition as compared withthe conventional calculation apparatus.

Since the FFT is carried out at N/2 points, y(m, n) is obtained in acase where k is variable between 0 and N/2−1, both inclusive. InEquation (24), p and k are symmetrical with one another. Using thesymmetric nature of the cosine with respect to k, Equation (23) isrewritten into: $\begin{matrix}\begin{matrix}{{y\left( {m,k} \right)} = {\sum\limits_{n = 0}^{N - 1}\quad{{{x2}(n)}{\cos\quad\left\lbrack {2{\pi\left( {k + {1\text{/}2}} \right)}\left( {n + {1\text{/}2}} \right)\text{/}N} \right\rbrack}}}} \\{= {\sum\limits_{n = 0}^{N - 1}\quad{{- {{x2}(n)}}{\cos\quad\left\lbrack {2{\pi\left( {\left( {N - 1 - k} \right) + {1\text{/}2}} \right)}\left( {n + {1\text{/}2}} \right)\text{/}N} \right\rbrack}}}} \\{= {- {{y\left( {m,{N - 1 - k}} \right)}.}}}\end{matrix} & (32)\end{matrix}$Therefore, y(m, k) is obtained in another case where k is variablebetween N/2 and N−1, both inclusive.

The description will proceed to another example of the algorithm that isapplicable to the inverse transform calculation apparatus according tothis invention. Substituting Equation (3) with p substituted for n,Equation (6) is rewritten into: $\begin{matrix}{{{xt}\left( {m,n} \right)} = {{2\text{/}N{\sum\limits_{k - 0}^{N - 1}{{y\left( {m,k} \right)}{\cos\left\lbrack {2{\pi\left( {p + {p\quad 0}} \right)}\left( {k + {1\text{/}2}} \right)\text{/}N} \right\rbrack}}}}\quad = {2\text{/}N{\sum\limits_{k = 0}^{N - 1}{{y\left( {m,k} \right)}{{\cos\left\lbrack {2{\pi\left( {p + {N\text{/}4} + {1\text{/}2}} \right)}\left( {k + {1\text{/}2}} \right)\text{/}N} \right\rbrack}.}}}}}} & (33)\end{matrix}$

For convenience of the equation, it will be assumed that:xt2(m, p)=xt(m, p−N/4).  (34)

In this event: $\begin{matrix}{{{xt2}\left( {m,p} \right)} = {{2\text{/}N{\sum\limits_{k = 0}^{N - 1}{{y\left( {m,k} \right)}{\cos\left\lbrack {2{\pi\left( {p + {1\text{/}2}} \right)}\left( {k + {1\text{/}2}} \right)\text{/}N} \right\rbrack}}}}\quad = \quad{{2\text{/}{N\left\lbrack {{\sum\limits_{k = 0}^{{N\text{/}2} - 1}{{y\left( {m,{2k}} \right)}{\cos\left( {2{\pi\left( {p + {1\text{/}2}} \right)}\left( {{2k} + {1\text{/}2}} \right)\text{/}N} \right)}}} + {\sum\limits_{k = 0}^{{N\text{/}2} - 1}{{y\left( {m,{N - 1 - {2k}}} \right)} \times {\cos\left( {2{\pi\left( {p + {1\text{/}2}} \right)}\left( {N - 1 - {2k} + {1\text{/}2}} \right)\text{/}N} \right)}}}} \right\rbrack}}\quad\quad = {2\text{/}{{N\left\lbrack {\sum\limits_{k = 0}^{{N\text{/}2} - 1}{\left( {{y\left( {m,{2k}} \right)} - {y\left( {m,{N - 1 - {2k}}} \right)}} \right) \times {\cos\left( {2{\pi\left( {p + {1\text{/}2}} \right)}\left( {{2k} + {1\text{/}2}} \right)\text{/}N} \right)}}} \right\rbrack}.}}}}} & (35)\end{matrix}$Substituting Equation (31): $\begin{matrix}{{{{xt}2}\left( {m,p} \right)} = {2\text{/}{{N\left\lbrack {\sum\limits_{k = 0}^{{N\text{/}2} - 1}{2{y\left( {m,{2k}} \right)}{\cos\left( {2{\pi\left( {p + {1\text{/}2}} \right)}\left( {{2k} + {1\text{/}2}} \right)\text{/}N} \right)}}} \right\rbrack}.}}} & (36)\end{matrix}$

Herein, y2(m, k) is represented as follows:y2(m, k)=y(m, 2k),  (37a)when 0≦k<N/4y2(m, k)=−y(m, k),  (37b)when N/4≦k<N/2.

Using Equation (32), Equation (37b) is represented as follows:y2(m, k)=−(m, N−1−2k),  (37c)when N/4≦k<N/2.

Equation (35) is rewritten into: $\begin{matrix}{{{{xt}2}\left( {m,p} \right)} = {{4\text{/}{{N{real}}\left\lbrack \quad{\sum\limits_{k = 0}^{{N\text{/}2} - 1}\quad{y\text{/}2\left( {m,k} \right) \times {\exp\left( {2\pi\quad{j\left( {p + {1\text{/}2}} \right)}\left( {{2k} + {1\text{/}2}} \right)\text{/}N} \right)}}} \right\rbrack}}\quad\quad = {{4\text{/}N{{real}\left\lbrack {\sum\limits_{k = 0}^{{N\text{/}2} - 1}\quad{{{y2}\left( {m,k} \right)} \times {\exp\left( {{2\pi\quad{j\left( {p + {1\text{/}2}} \right)}\text{/}2N} + {2\pi\quad{j\left( {p + {1\text{/}2}} \right)}2k\text{/}N}} \right)}}} \right\rbrack}}\quad = {{4\text{/}N{{real}\left\lbrack {{\exp\left( {2\pi\quad{j\left( {p + {1\text{/}2}} \right)}\text{/}2N} \right)} \times {\sum\limits_{k = 0}^{{N\text{/}2} - 1}\quad{{{y2}\left( {m,k} \right)}{\exp\left( {2\pi\quad{j\left( {{2p} + 1} \right)}\quad k\text{/}N} \right)}}}} \right\rbrack}}\text{}\quad = {4\text{/}N{{real}\left\lbrack {{\exp\left( {2\pi\quad{j\left( {p + {1\text{/}2}} \right)}\text{/}2N} \right)} \times {\sum\limits_{k = 0}^{{N\text{/}2} - 1}\quad{{{y2}\left( {m,k} \right)}{\exp\left( {{2\pi\quad{jk}\text{/}N} + \left( {2\pi\quad{jpk}\text{/}\left( {N\text{/}2} \right)} \right)} \right\rbrack}}}} \right.}}}}}} & (38)\end{matrix}$It is assumed that:

y3(m, k)=y2(m, k)exp(2πjk/N)  (39) $\begin{matrix}{{{y4}\left( {m,p} \right)} = {\sum\limits_{k = 0}^{{N\text{/}2} - 1}\quad{{{y3}\left( {m,k} \right)}{\exp\left\lbrack {2\pi\quad{jpk}\text{/}\left( {N\text{/}2} \right)} \right\rbrack}}}} & (40)\end{matrix}$

Equation (40) represents an inverse FFT for y3(m, k). SubstitutingEquations (39) and (40) into Equation (38):xt2(m, n)=4/N real[exp(2πj(n+1/2)/2N)y4(m, n)].  (41)

In order to convert xt2(m, n) to xt(m, n), it is assumed from Equations(20), (24), and (32) that:xt(m, 3N/4−1−n)=xt(m, 3N/4+n)  (42a)=−xt2(m, n)when 0≦n<N/4It is therefore possible to obtain the result of Equation (6).

The multiplication is carried out the number of times which issubstantially equal to (N/2) log₂ (N/2). The addition is carried out thenumber of times which is substantially equal to Nlog₂ (N/2).Accordingly, it is possible to reduce the number of times of themultiplication and the addition as compared with the conventionalcalculation apparatus.

Referring to FIG. 2, the description will be directed to a calculationapparatus according to a first embodiment of this invention. Thecalculation apparatus comprises similar parts designated by likereference numerals. The forward transform window part 14 produces, asthe product signal, a succession of zeroth through (3N/4−1)th and(3N/4)th through (N−1)th product data. The forward transform window part14 will be referred to as a multiplying arrangement.

The forward transform unit 11a further comprises first forwardprocessing, second forward processing, and forward FFT parts 21, 22, and23. The first forward processing part 21 is connected to the forwardtransform window part 14 and is for processing the product signal into aprocessed signal.

Referring to FIG. 3 together with FIG. 2, the description will be madeas regards operation of the first forward processing part 21. At a firststage SA1, the first forward processing part 21 is supplied with theproduct signal from the forward transform window part 14. The firststage SA1 proceeds to a second stage SA2 at which the first forwardprocessing part 21 processes the (3N/4)th through the (N−1)th productdata into a succession of zeroth through (N/4−1)th particular datahaving a negative polarity or sign in common. In other words, the zeroththrough the (N/4−1)th product data are processed in accordance withEquation (11a). The first forward processing part 21 for carrying outthe second stage SA2 will be referred to as a particular processingarrangement.

The second stage SA2 proceeds to a third stage SA3 at which the firstforward processing part 21 processes the zeroth through the (3N/4−1)thproduct data into a succession of (N/4)th through (3N/4−1)th specificdata having a positive polarity or sign in common. In other words, the(N/4)th through the (N−1)th product data are processed in accordancewith Equation (11b). The first forward processing part 21 for carryingout the third stage SA3 will be referred to as a specific processingarrangement.

The third stage SA3 proceeds to a fourth stage SA4 at which the firstforward processing part 21 multiplies exp[−2πjn/(2n)] and each of the(3N/4)through the N-th particular and the zeroth through the (3N/4−1)thspecific data in accordance with Equation (13) to produce the processedsignal. The first forward processing part 21 for carrying out the fourthstage SA4 will be referred to as a calculating arrangement.

Returning to FIG. 2, the forward FFT part 23 is connected to the firstforward processing part 21 and is for carrying out a linear forward FFTon the processed signal in accordance with Equation (15) to produce aninternal signal representative of a result of the forward FFT. Theforward FFT part 23 is herein referred to as an internal transformcarrying out arrangement.

The second forward processing part 22 is connected to the forward FFTpart 23 and is for processing the internal signal into the forwardtransformed signal in accordance with Equation (16). More particularly,the second forward processing part 22 multiplies the internal signal and[−2πj(k+½)2N] into a local product, namely, a real part, to make theforward transformed signal represent the local product. In this event,the second forward processing part 22 will be referred to as an internalmultiplying arrangement. A combination of the first forward processing,the second forward processing, and the forward FFT parts 21, 22, and 23will be referred to as a transform carrying out arrangement.

Continuing reference to FIG. 2, the description will proceed to theinverse transform unit 12a. The inverse transform unit 12 comprisesfirst inverse processing, second inverse processing, and inverse FFTparts 31, 32, and 33. The first inverse processing part 31 is connectedto the second forward processing part 22 and is for processing theforward transformed signal into a processed signal.

The first inverse processing part 31 is supplied with the forwardtransformed signal as an apparatus input signal. The forward transformedsignal is a succession of zeroth through (N−1)th apparatus input data.

The first inverse processing part 31 carries out multiplication betweenthe zeroth through the (N−1)th apparatus input data andexp[2πj(N/4+½)k/N] in accordance with Equation (18) into a first productto make the processed signal represent the first product. In this event,the first inverse processing part 31 will be referred to as a firstmultiplying arrangement.

The inverse FFT part 33 is connected to the first inverse processingpart 31 and is for carrying out a linear inverse FFT on the processedsignal in accordance with Equation (19) to produce an internal signalrepresentative of a result of the inverse FFT. The internal signal is asuccession of zeroth through (N−1)th internal data. In this event, theinverse FFT part 33 is herein referred to as an internal transformcarrying out arrangement.

The second inverse processing part 32 is connected to the inverse FFTpart 33 and is for carrying out multiplication between the zeroththrough the (N−1)th internal data and 2exp[−2πj(n+N/4+½)/(2N)] inaccordance with Equation (20) into a second product, namely, a realpart, to make the inverse transformed signal represent the secondproduct. In this event, the second inverse processing part 32 will bereferred to as a second multiplying arrangement.

The second inverse processing part 32 is further connected to theinverse transform window part 17. The inverse transformed signal is sentfrom the second inverse processing part 32 to the inverse transformwindow part 17.

Referring to FIG. 4, the description will be directed to a calculationapparatus according to a second embodiment of this invention. Thecalculation apparatus comprises similar parts designated by likereference numerals. The forward transform window part 14 produces, asthe product signal, a succession of zeroth through (3N/4−1)th and(3N/4)th through (N−1)th product data. The forward transform window part14 will be referred to as a multiplying arrangement.

The first forward processing part 21 comprises subtracting andmultiplying parts 41 and 42. The subtracting part 41 is connected to theforward transform window part 14 and is for producing a local signal inresponse to the product signal. The multiplying part 42 is connected tothe subtracting part 41 and is for producing the processed signal inresponse to the local signal.

Referring to FIG. 5 together with FIG. 4, the description will be madeas regards operation of the first forward processing part 21. At a firststage SB1, the subtracting part 41 is supplied with the product signalfrom the forward transform window part 14. The first stage SB1 proceedsto a second stage SB2 at which the subtracting part 41 processes thezeroth through the (N/4−1)th product data into a succession of (3N/4)through Nth particular data having a negative polarity in common. Inother words, the zeroth through the (N/4−1)th product data are processedin accordance with Equation (22a). The subtracting part 41 for carryingout the second stage SB2 will be referred to as a particular processingarrangement.

The second stage SB2 proceeds to a third stage SB3 at which thesubtracting part 41 processes the zeroth through the (3N/4−1)th productdata into a succession of zeroth through (N/4)th specific data having apositive polarity in common. In other words, the (N/4)th through the(N−1)th product data are processed in accordance with Equation (22b).The subtracting part 41 for carrying out the third stage SB3 will bereferred to as a specific processing arrangement.

The third stage SB3 proceeds to a fourth stage SB4 at which thesubtracting part 41 combines the particular and the specific datasuccessions into a succession of zeroth through (N−1−2p)th and 2pththrough (N−1)th combined data. When the fourth stage SB3 is carried out,the subtracting part 41 will be referred to as a combining arrangement.

The fourth stage SB4 proceeds to a fifth stage SB5 at which thesubtracting part 41 subtracts the (N−1−2p)th combined datum from the2pth combined datum in accordance with Equation (26) to produce adifference and the local signal that is representative of thedifference. The subtracting part 41 for carrying out the fifth stage SB5will be referred to as a subtracting arrangement.

The fifth stage SB5 proceeds to a sixth stage SB6 at which themultiplying part 42 carries out multiplication between exp(−2πjp/N) andthe local signal in accordance with Equation (28) to produce an internalproduct to make the processed signal represent the internal product. Themultiplying part 42 will be referred to as an internal multiplyingarrangement.

Returning to FIG. 4, the forward FFT part 23 is connected to themultiplying part 42 and is for carrying out the linear forward FFT onthe processed signal in accordance with Equation (30) to produce aninternal signal representative of a result of the forward FFT. Theforward FFT part 23 is herein referred to as an internal transformcarrying out arrangement.

The second forward processing part 22 is connected to the forward FFTpart 23 and is for processing the internal signal into the forwardtransformed signal in accordance with Equation (31). More particularly,the second forward processing part 22 carries out multiplication betweenthe internal signal and [−2πj(k+½)/(2N)] into a local product, namely, areal part, to make the forward transformed signal represent the localproduct. In this event, the second forward processing part 22 will bereferred to as an internal multiplying arrangement. A combination of thefirst forward processing, the second forward processing, and the forwardFFT parts 21, 22, and 23 will be referred to as a transform carrying outarrangement.

Referring to FIG. 6 together with FIG. 4, operation of the first inverseprocessing part 31 will be described at first. At a first stage SC1, thefirst inverse processing part 31 is supplied with the forwardtransformed signal from the second forward processing part 22. The firststage SC1 proceeds to a second stage SC2 at which the first inverseprocessing part 31 processes the 2kth apparatus input datum into a kthparticular datum. In other words, the forward transformed signal isprocessed in accordance with Equation (37a). In this event, the firstinverse processing part 16 will be referred to as a particularprocessing arrangement.

The second stage SC2 proceeds to a third stage SC3 at which the firstinverse processing part 31 processes the (2k+1)th apparatus input datuminto a (N−1−k)th specific datum. In other words, the forward transformedsignal is processed in accordance with Equation (37c). In this event,the first inverse processing part 16 will be referred to as a specificprocessing arrangement.

The third stage SC3 proceeds to a fourth stage SC4 at which the firstinverse processing part 31 carries out multiplication betweenexp(2njk/N) and each of the kth particular and the (N−1−k)th specificdata in accordance with Equation (38) into the processed signal. In thisevent, the first inverse processing part 16 will be referred to as acalculation arrangement.

In FIG. 4, the inverse FFT part 33 carries out the linear inverse FFT onthe processed signal in accordance with Equation (40) to produce aninternal signal representative of a result of the inverse FFT. Theinverse FFT part 23 will be referred to as an internal transformcarrying out arrangement.

Referring to FIG. 7 together with FIG. 4, the description will bedirected to operation of the second inverse processing part 32. Theinverse FFT part 23 produces, as the internal signal, a succession ofzeroth through (p−1)th and pth through (N/2−1)th internal data. At afirst stage SD1, the second inverse processing part 32 is supplied withthe internal signal from the inverse FFT part 23.

The first stage SD1 proceeds to a second stage SD2 at which the secondinverse processing part 32 carries out multiplication between the pthinternal datum and the exp[2πj(p+½)/(2N)] in accordance with Equation(41) into a local product to make the inverse transformed signalrepresent the local product. The local product is a succession of zeroththrough (N/4−1)th and (N/4)th through (N/2−1)th product data. In thisevent, the second inverse processing part 32 will be referred to as amultiplying arrangement.

The second stage SD2 proceeds to a third stage SD3 at which the secondinverse processing part 32 processes the zeroth through the (N/4−1)thproduct data in accordance with Equation (42a) into a first successionof (3N/4−1)th through (N/2)th particular data in a descending order anda second succession of (3N/4)th through Nth particular data in ascendingorder. The particular data of the first and the second successions havea first polarity in common. In this event, the second inverse processingpart 32 will be referred to as a particular processing arrangement.

The third stage SD3 proceeds to a fourth stage SD4 at which the secondinverse processing part 32 processes the (N/4)th through the (N/2−1)thproduct data in accordance with Equation (42b) into a first successionof zeroth through (N/4−1)th specific data in an ascending order and asecond succession of (N/2−1)th through (N/4)th specific data in adescending order. The specific data of the first and the secondsuccessions have a second polarity in common. The second polarity isdifferent from the first polarity. In this event, the second inverseprocessing part 32 will be referred to as a specific processingarrangement.

While the present invention has thus far been described in connectionwith only a few embodiment thereof, it will readily be possible forthose skilled in the art to put this invention into practice in variousother manners. For example, the block length N may be equal to 256 or512.

1. An apparatus for carrying out a forward modified discrete cosinetransform comprising: an input signal; a multiplier, said multipliermultiplying a predetermined forward transform window function and saidinput signal and outputting as a result a product signal; transformcarrying out means connected to said multiplier for carrying out alinear forward modified discrete cosine transform on said product signaland for outputting a forward modified discrete cosine transformed signalrepresentative of said linear forward modified discrete cosinetransform, wherein said transform carrying out means further comprises:a first processing device connected to receive said product signal fromsaid multiplier, said first processing device outputting a processedsignal; means connected to said first processing device for receivingsaid processed signal and carrying out a forward fast Fourier transformon said processed signal and outputting an internal signalrepresentative of said forward fast Fourier transform; and a secondprocessing device connected to receive said internal signal from saidmeans for carrying out a forward fast Fourier transform, said secondprocessing device processing said internal signal and outputting as aresult said forward modified discrete cosine transformed signal.
 2. Anapparatus as recited in claim 1, wherein said product signal produced bysaid multiplier, is a succession of zeroth through (N/4−1)th and (N/4)ththrough (N−1)th product data, where N represents an integral multiple offour; said first processing device includes a particular processingmeans connected to said multiplier for processing said zeroth throughsaid (N/4−1)th product data into a succession of (3N/4) through Nthparticular data having a first polarity in common; said first processingdevice further includes a specific processing means connected to saidmultiplier for processing said (N/4)th through said (N−1)th product datainto a succession of zeroth through (3N/4−1)th specific data having asecond polarity in common, said second polarity being different fromsaid first polarity; and a calculating means is connected to saidparticular processing means, said specific processing means, and saidmeans for carrying out a forward fast Fourier transform, for calculatingsaid processed signal by using a predetermined signal and each of said(3N/4) through said Nth particular and said zeroth through said(3N/4−1)th specific data.
 3. An apparatus as recited in claim 2, whereinsaid predetermined signal represents exp(−2πjn/(2N)), and saidcalculating means multiplies said exp(−2πjn/(2N)) and each of said(3N/4) through said Nth particular data and said zeroth through the(3N/4−1)th specific data to produce said processed signal, where jrepresents an imaginary unit, n being variable between 0 and N−1, bothinclusive.
 4. An apparatus as recited in claim 2, wherein saidcalculating means comprises: combining means connected to saidparticular and said specific processing means for combining saidparticular and said specific data successions into a succession ofzeroth through (N−1−2p)th and 2pth through (N−1)th combined data, wherep is variable between 0 and N/2−1, both inclusive; a subtractorconnected to said combining means, said subtractor subtracting said(N−1−2p)th combined datum from said 2pth combined datum to produce adifference and output a local signal representative of said difference;and internal multiplying means connected to said subtractor and saidmeans for carrying out a forward fast Fourier transform, for multiplyinga predetermined signal with said local signal into an internal productto make said processed signal represent said internal product.
 5. Anapparatus as recited in claim 4, wherein said predetermined signalrepresents exp(−2πjp/N), and said internal multiplying means multiplessaid exp(−2πjp/N) and said local signal to produce said processedsignal, where j represents an imaginary unit, p being variable between 0and N−1, both inclusive.
 6. An apparatus as recited in claim 1, whereinsaid internal signal is a succession of zeroth through (K−1)th and kththrough (N/2−1)th internal data, where N represents an integral multipleof four, k being variable between 0 and N−1, both inclusive, and whereinsaid second processing device includes internal multiplying meansconnected to said means for carrying out a forward fast Fouriertransform, for multiplying said kth internal datum andexp(−2πj(k+½)/(2N)) into a local product to make said forwardtransformed signal represent said local product, where j represents animaginary number.
 7. An apparatus for carrying out an inverse modifieddiscrete cosine transform comprising: an input signal comprising amodified discrete cosine transformed signal; transform carrying outmeans for carrying out a linear inverse modified discrete cosinetransform on said input signal and for outputting an inverse modifieddiscrete cosine transformed signal representative of a result of saidlinear inverse modified discrete cosine transform; a multiplierconnected to said transform carrying out means, said multipliermultiplying a predetermined inverse transform window function and saidinverse modified discrete cosine transformed signal to produce a productsignal; wherein said transform carrying out means comprises: a firstprocessing device which receives said input signal and outputs aprocessed signal; internal transform carrying out means connected tosaid first processing device for carrying out an inverse fast Fouriertransform on said processed signal and for outputting as a result ofsaid inverse fast Fourier transform an internal signal; and a secondprocessing device connected to said internal transform carrying outmeans to receive said internal signal and output as a result ofprocessing said internal signal said inverse modified discrete cosinetransformed signal.
 8. An apparatus as recited in claim 7, said inputsignal being a succession of zeroth through (N−1)th apparatus inputdata, where N represents an integral multiple of four, wherein saidfirst processing device includes a first multiplier, said multipliermultiplying said zeroth through said (N−1)th apparatus input data andexp(2π(N/4+½)k/N) and outputting as a result a first product, saidprocessed signal representing said first product, where j represents animaginary unit, k being variable between 0 and N−1, both inclusive. 9.An apparatus as recited in claim 7, said internal transform carrying outmeans producing, as said internal signal, a succession of zeroth through(N−1)th internal data, where N represents an integral multiple of four,wherein said second processing device includes a second multiplierconnected to said internal transform carrying out means, said multipliermultiplying said zeroth through said (N−1)th internal data andexp(−2πj(n+N/4+½)/(2N)) into a second product, said inverse transformedsignal representing said second product, where j represents an imaginaryunit, n being variable between 0 and N−1, both inclusive.
 10. Anapparatus as recited in claim 7, wherein said input signal is asuccession of zeroth through (N/2−1)th apparatus input data, where Nrepresents an integral multiple of four; said first processing deviceincludes a particular processing means for processing said 2kthapparatus input datum into a kth particular datum, where k is variablebetween 0 and N/2−1, both inclusive and a specific processing means forprocessing said (2k+1)th apparatus input datum into a (N−1−k)th specificdatum; and a calculating means connected to said particular and saidspecific processing means for calculating said processed signal by usinga predetermined signal and each of said kth particular and said(N−1−k)th specific data.
 11. An apparatus as recited in claim 10,wherein said predetermined signal represents exp(2πjk/N), where jrepresents an imaginary unit, and said calculating means multiplies saidpredetermined signal and said kth particular datum.
 12. An apparatus asrecited in claim 7, said internal transform carrying out meansproducing, as said internal signal, a succession of zeroth through(p−1)th and pth through (N/2−1)th internal data, where N represents anintegral multiple of four, p being variable between 0 and (N/2−1), bothinclusive, wherein said second processing device comprises: a multiplierconnected to said internal transform carrying out means, said multipliermultiplying said pth internal datum and exp(2πj(p+½)/2N) resulting in alocal product to make said inverse transformed signal represent saidlocal product, j representing an imaginary unit, said local productbeing a succession of zeroth through (N/4−1)th and (N/4)th through(N/2−1)th product data; a particular processing means connected to saidmultiplier for processing said zeroth through said (N/4−1)th productdata into a first succession of (3/N4−1) ( 3N/4−1 )th through (N/2)thparticular data in a descending order and a second succession of(3N/4)th through Nth particular data in an ascending order, saidparticular data of said first and said second successions having a firstpolarity in common; and a specific processing means connected to saidmultiplier for processing said (N/4)th through (N/2−1)th product datainto a first succession of zeroth through (N/4−1)th specific data in anascending order and a second succession of (N/2−1)th through (N/4)thspecific data in a descending order, the specific data of said first andsaid second successions having a second polarity in common, said secondpolarity being different from said first polarity.
 13. The apparatus ofclaim 1 wherein said input signal is an audio signal.
 14. The apparatusof claim 1 wherein said modified discrete cosine transformed signal hasa block length N of
 512. 15. The apparatus of claim 1 wherein saidtransform carrying out means calculates fewer than N² multiplications.16. The apparatus of claim 1 wherein said transform carrying out meanscalculates fewer than N(N−1 ) additions.
 17. The apparatus of claim 1wherein said processed signal is formed by multiplying said input signalby a predetermined factor.
 18. The apparatus of claim 17 wherein saidmodified discrete cosine transformed signal comprises said internalsignal representative of said forward fast Fourier transform multipliedby a second predetermined factor.
 19. The apparatus of claim 7 whereinsaid modified discrete cosine transformed signal is a transformed audiosignal.
 20. The apparatus of claim 7 wherein said modified discretetransformed signal has a block length N of
 512. 21. The apparatus ofclaim 7 wherein said transform carrying out means calculates fewer thanN² multiplications to carry out said inverse discrete cosine transform.22. The apparatus of claim 21 wherein said transform carrying out meanscalculates fewer than N(N−1 ) additions to carry out said inversediscrete cosine transform.
 23. The apparatus of claim 7 wherein saidprocessed signal comprises said input signal multiplied by apredetermined factor.
 24. The apparatus of claim 23 wherein said inversemodified discrete cosine transform signal comprises said internal signalrepresentative of said inverse forward fast Fourier transform multipliedby a second predetermined factor.
 25. An apparatus for carrying out aninverse transform comprising: an input signal y(m,k); transform carryingout means for carrying out a linear inverse transform xt(m,n) on saidinput signal y(m,k) and for outputting an inverse transformed signalrepresentative of a result of said linear inverse transform, said linearinverse transform being defined by:${{xt}\left( {m,n} \right)} = {2\text{/}N{\sum\limits_{k = 0}^{N - 1}{{y\left( {m,k} \right)}{\cos\left\lbrack {2{\pi\left( {n + {n0}} \right)}\left( {k + {1\text{/}2}} \right)\text{/}N} \right\rbrack}}}}$where m represents a block number, n represents a sample number, Nrepresents a block length and k is an integer between 0 and N−1; amultiplier connected to said transform carrying out means, saidmultiplier multiplying a predetermined inverse transform window functionand said inverse transformed signal to produce a product signal; whereinsaid transform carrying out means comprises: a first processing devicewhich receives said input signal y(m,k) and outputs a processed signal,said processed signal comprising a product signal formed by multiplyingsaid input signal y(m,k) by a predetermined factor; internal transformcarrying out means connected to said first processing device forcarrying out an inverse fast Fourier transform on said processed signaland for outputting as a result of said inverse fast Fourier transform aninternal signal; and a second processing device connected to saidinternal transform carrying out means to receive said internal signaland output as a result of processing said internal signal said inversetransformed signal.
 26. The apparatus of claim 17 wherein N is
 512. 27.The apparatus of claim 17 wherein said transform carrying out meanscalculates fewer than N² multiplications.
 28. The apparatus of claim 19wherein said transform carrying out means calculates fewer than N(N−1 )additions.
 29. An apparatus for carrying out an inverse transformcomprising: a transformed discrete audio input signal having a blocksize (N) of 512; transform carrying out means for carrying out a linearinverse transform on said input signal by calculating fewer than N ²multiplications and fewer than N(N−1 ) additions, and for outputting aninverse transformed signal representative of a result of said linearinverse transform; a multiplier connected to said transform carrying outmeans, said multiplier multiplying a predetermined inverse transformwindow function and said inverse transformed signal to produce a productsignal; wherein said transform carrying out means comprises: a firstprocessing device which receives said input signal and outputs aprocessed signal; internal transform carrying out means connected tosaid first processing device for carrying out an inverse fast Fouriertransform on said processed signal and for outputting as a result ofsaid inverse fast Fourier transform an internal signal; and a secondprocessing device connected to said internal transform carrying outmeans to receive said internal signal and output as a result ofprocessing said internal signal said inverse transformed signal.
 30. Anapparatus for carrying out a forward transform comprising: an inputsignal x(n); a multiplier for multiplying a predetermined forwardtransform window function and said input signal x(n) to produce aproduct signal to produce a product signal xh(n); transform carrying outmeans connected to said multiplier for carrying out a linear forwardtransform on said product signal xh(n) and for outputting a forwardtransformed signal y(m,k) representative of a result of said forwardtransform, said linear forward transform being defined by;${y\left( {m,k} \right)} = {\sum\limits_{n = 0}^{N - 1}\quad{{{xh}(n)}{\cos\quad\left\lbrack {2{\pi\left( {k + {1\text{/}2}} \right)}\left( {n + {n\quad 0}} \right)\text{/}N} \right\rbrack}}}$where m represents a block number, n represents a sample number, Nrepresents a block length and k is an integer between 0 and N−1; whereinsaid transform carrying out means comprises: a first processing devicewhich receives said product signal xh(n) and outputs a processed signal,said processed signal comprising a product signal formed by multiplyingsaid product signal xh(n) by a predetermined factor; internal transformcarrying out means connected to said first processing device forcarrying out a forward fast Fourier transform on said processed signaland for outputting as a result of said forward fast Fourier transform aninternal signal; and a second processing device connected to saidinternal transform carrying out means to receive said internal signaland output as a result of processing said internal signal said forwardtransformed signal.
 31. An apparatus for carrying out a forwardtransform comprising: an input signal having a block size (N) of 512; amultiplier for multiplying a predetermined forward transform windowfunction and said input signal to produce a product signal; andtransform carrying out means connected to said multiplier for carryingout a linear forward transform on said product signal by calculatingfewer than N ² multiplications and fewer than N(N−1 ) additions, and foroutputting a forward transformed signal representative of a result ofsaid linear forward transform; wherein said transform carrying out meanscomprises: a first processing device which receives said product signaland outputs a processed signal; internal transform carrying out meansconnected to said first processing device for carrying out a forwardfast Fourier transform on said processed signal and for outputting as aresult of said forward fast Fourier transform an internal signal; and asecond processing device connected to said internal transform carryingout means to receive said internal signal and output as a result ofprocessing said internal signal said forward transformed signal.
 32. Anapparatus for carrying out a forward modified discrete cosine transformcomprising: an input signal; a multiplier, said multiplier multiplying apredetermined forward transform window function and said input signaland outputting as a result a product signal; and transform carrying outmeans connected to said multiplier for carrying out a linear forwardmodified discrete cosine transform on said product signal and foroutputting a forward modified discrete cosine transformed signalrepresentative of said linear forward modified discrete cosinetransform, wherein said transform carrying out means comprises: a firstprocessing device connected to receive said product signal having Nsamples, N being an integer, from said multiplier, said first processingdevice outputting a processed signal having M samples, M being aninteger different from N; means connected to said first processingdevice for receiving said processed signal and for carrying out aforward fast Fourier transform on said processed signal, and foroutputting an internal signal representative of said forward fastFourier transform; and a second processing device connected to receivesaid internal signal from said means for carrying out a forward fastFourier transform, said second processing device processing saidinternal signal and outputting as a result said forward modifieddiscrete cosine transformed signal.
 33. The apparatus as claimed inclaim 32, wherein M is smaller than N.
 34. The apparatus as claimed inclaim 32, wherein M is equal to N/2.
 35. An apparatus for carrying outan inverse modified discrete cosine transform comprising: an inputsignal having M samples, M being an integer; transform carrying outmeans carrying out a linear inverse modified discrete cosine transformon said input signal and for outputting an inverse modified discretecosine transformed signal having M samples representative of said linearinverse modified discrete cosine transform; and a multiplier connectedto said transform carrying out means, said multiplier multiplying apredetermined inverse transform window function and said linear inversemodified discrete cosine transformed signal to produce a product signalhaving N samples, N being an integer different from M, wherein saidtransform carrying out means comprises: a first processing device whichreceives said input signal, said first processing device outputting aprocessed signal; internal transform carrying out means connected tosaid first processing device for carrying out an inverse fast Fouriertransform on said processed signal as a result of processing saidinverse fast Fourier transform an internal signal; and a secondprocessing device connected to said internal transform carrying outmeans to receive said internal signal and output as a result ofprocessing said inverse modified discrete cosine transformed signal. 36.The apparatus as claimed in claim 35, wherein M is smaller than N. 37.The apparatus as claimed in claim 35, wherein M is equal to N/2.
 38. Anapparatus for carrying out a forward transform comprising: an inputsignal x(n); a multiplier for multiplying a predetermined forwardtransform window function and said input signal x(n) to produce aproduct signal xh(n); and transform carrying out means connected to saidmultiplier for carrying out a linear forward transform on said productsignal xh(n) and for outputting a forward transformed signal y(m,k)representative of a result of said forward transform, said linearforward transform being defined by;${y\left( {m,k} \right)} = {\sum\limits_{n = 0}^{N - 1}\quad{{{xh}(n)}{\cos\quad\left\lbrack {2{\pi\left( {k + {1\text{/}2}} \right)}\left( {n + {n\quad 0}} \right)\text{/}N} \right\rbrack}}}$where m represents a block number, n represents a sample number, Nrepresents a block length and k is an integer between 0 and N−1; whereinsaid transform carrying out means comprises: a first processing devicewhich receives said product signal xh(n) having N samples and outputs aprocessed signal having M samples, M being an integer different from N,said processed signal comprising a product signal formed by multiplyingsaid product signal xh(n) by a predetermined factor; internal transformcarrying out means connected to said first processing device forcarrying out a forward fast Fourier transform on said processed signaland for outputting as a result of said forward fast Fourier transform aninternal signal; and a second processing device connected to saidinternal transform carrying out means to receive said internal signaland output as a result of processing said internal signal said forwardtransformed signal.
 39. The apparatus as claimed in claim 38, wherein Mis smaller than N.
 40. The apparatus as claimed in claim 38, wherein Mis equal to N/2.
 41. An apparatus for carrying out an inverse transformcomprising: an input signal y(m,k) having M samples, M being an integer;transform carrying out means for carrying out a linear inverse transformon said input signal y(m,k) and for outputting an inverse transformedsignal xt(m,n) representative of a result of said linear inversetransform, said linear inverse transform being defined by:${{xt}\left( {m,n} \right)} = {2\text{/}N{\sum\limits_{k = 0}^{N - 1}{{y\left( {m,k} \right)}{\cos\left\lbrack {2{\pi\left( {n + {n0}} \right)}\left( {k = {1\text{/}2}} \right)\text{/}N} \right\rbrack}}}}$where m represents a block number, n represents a sample number, Nrepresents a block length and k is an integer between 0 and N−1; amultiplier connected to said transform carrying out means, saidmultiplier multiplying a predetermined inverse transform window functionand said inverse transformed signal xt(m,n) to produce a product signalhaving N samples, N being an integer different from M; wherein saidtransform carrying out means comprises: a first processing device whichreceives said input signal y(m,k) and outputs a processed signal, saidprocessed signal comprising a product signal formed by multiplying saidinput signal y(m,k) by a predetermined factor; internal transformcarrying out means connected to said first processing device forcarrying out an inverse fast Fourier transform on said processed signaland for outputting as a result of said inverse fast Fourier transform aninternal signal; and a second processing device connected to saidinternal transform carrying out means to receive said internal signaland output as a result of processing said internal signal said inversetransformed signal.
 42. The apparatus as claimed in claim 41, wherein Mis smaller than N.
 43. The apparatus as claimed in claim 41, wherein Mis equal to N/2.
 44. An apparatus for carrying out an inverse transformcomprising: an input signal y(m,k) having M samples, M being an integer;transform carrying out means for carrying out a linear inverse transformon said input signal y(m,k) and for outputting an inverse transformedsignal xt(m,n) representative of a result of said linear inversetransform, said linear inverse transform xt(m,n) being determined usingthe sum:$\sum\limits_{k = 0}^{M - 1}{{y\left( {m,k} \right)}{\cos\left\lbrack {2{\pi\left( {n + {n0}} \right)}\left( {k = {1\text{/}2}} \right)\text{/}N} \right\rbrack}}$where m represents a block number, n represents a sample number, Nrepresents a block length and k is an integer between 0 and M−1; amultiplier connected to said transform carrying out means, saidmultiplier multiplying a predetermined inverse transform window functionand said inverse transformed signal xt(m,n) to produce a product signalhaving N samples, N being an integer different from M; wherein saidtransform carrying out means comprises: a first processing device whichreceives said input signal y(m,k) and outputs a processed signal, saidprocessed signal comprising a product signal formed by multiplying saidinput signal y(m,k) by a predetermined factor; internal transformcarrying out means connected to said first processing device forcarrying out an inverse fast Fourier transform on said processed signaland for outputting as a result of said inverse fast Fourier transform aninternal signal; and a second processing device connected to saidinternal transform carrying out means to receive said internal signaland output as a result of processing said internal signal said inversetransformed signal.
 45. The apparatus as claimed in claim 44, wherein Mis smaller than N.
 46. The apparatus as claimed in claim 44, wherein Mis equal to N/2.